Normat 53:4, 173–185 (2005) 173
Mathematics education and historical
references: Guido Grandi’s infinite series
Giorgio T. Bagni
Department of Mathematics and Computer Science
University of Udine
Via delle Science 206
IT-33100 Udine, Italy
bagni@dimi.uniud.it
Abstract
Integrating history of mathematics into the mathematics education is an im-
portant point: the use of history in education can be an effective tool for the
teacher. In this paper an example from the history of mathematics (Grandi’s
infinite series, 1703, and Leibniz remarks, 1716) is presented and its educa-
tional utility is investigated. Students’ behaviour is examined, with reference
to pupils aged 16–18 years: we conclude that historical examples are useful in
order to improve teaching of infinite s eries, e.g. to stimulate reflections about
actual and potential infinity. Nevertheless the main problem of the passage
from finite to infinite is a cultural one, and historical issues are important
in order to approach it, although the historical approach can be considered
together with other educational approaches.
Key words: actual and potential infinity, didactical contract, history of
mathematics, infinite series, probabilistic argum ent, socio-cultural context.
History and mathematical education
Even 500 years ago a philosophy of mathematics was possible,
a philosophy of what mathematics was then.
Ludwig Wittgenstein (1956, IV, 53)
Several theoretical frameworks can be mentioned in order to link learning processes
with historical issues (Fauvel & van Maanen, 2000; Cantoral & Farfán, 2004, see in
particular the Chapter 8). According to the “epistemological obstacles” perspective
(Bachelard, 1938; Brousseau, 1983), one of the main goals of historical study is
finding systems of constraints (the so-called situations fondamentales) that must
be studied in order to understand knowledge, whose discovery is connected to
their solution. Some Authors (Radford, Boero & Vasco 2000, p. 163) notice that
this perspective is characterised by an important assumption: the reappearance
in our teaching-learning processes, in the present, of the obstacles encountered
by mathematicians in the past. Nevertheless, historical data must be considered
nowadays and several issues are connected with their interpretation, based upon
174 Giorgio T. Bagni Normat 4/2005
our cultural institutions and beliefs; according to Luis Radford’s socio-cultural
perspective, knowledge is linked to activities of individuals and this is essentially
related to cultural institutions (Radford, 1997); knowledge is not built individually,
but in a wider social context (Radford, Boero & Vasco, 2000, p. 164). So the
savoir savant (Chevallard, 1985) cannot be considered absolute and it must be
understood in terms of cultural institutions (Lizcano, 1993; Barbin, 1994; Grugnetti
& Rogers, 2000; Dauben & Scriba, 2002). A first notion of infinite s eries may well
have a very ancient source: Aristotle of Stagira (384–322 BC) implicitly underlined
that the sum of a series of infinitely many addends (potentially considered) can
be a finite quantity (Physics, III, VI, 206 b, 1–33). In his Quadratura parabolae,
Archimedes of Syracuse (287–212 BC) considered implicitly a geometric series.
Several centuries later, Andreas Tacquet (1612–1660) noticed that the passage from
a “finite progression” to an infinite series would be “immediate” (Loria, 1929–1933,
p. 517); but such a passage is crucial:
1
Greek conceptions strictly distinguished
actual and potential infinity (and mathematical infinity, following Aristotle, was
accepted only in a potential sense).
2
In this work we shall discuss some educational
aspects related to infinite series by using historical examples (see for instance:
Edwards, 1994; Hairer & Wanner, 1996; Bagni, 2000a). In particular, we are going
to discuss two main points:
Can we consider effectively a parallelism between students’ justifications and
some examples from the history of mathematics? And does this bear out the well-
known idea by Piaget and Garcia (see: Piaget & Garcia, 1983), according to which
historical development and individual development are linked? Moreover, is our
students’ approach to infinite series influenced by potential infinity or by actual
infinity?
We shall consider a brief historical survey and a case study,
3
in order to propose
some hints for further research.
Grandi, Leibniz, Riccati
Grandi and Leibniz
We are going to examine a well-known indeterminate series. In 1703, Guido Grandi
(1671–1742) noticed that from 1 1+1 1+··· it is possible to obtain 0 or 1:
(1 1) + (1 1) + (1 1) + (1 1) + ···=0+0+0+0+···=0
1+(1 + 1) + (1 + 1) + (1 + 1) + ··· =1+0+0+0+··· =1
The sum of the alternating series 1 1+1 1+··· was considered
1
2
by Grandi.
4
According to him, the proof can be based upon the following expansion (now ex-
pressed using modern notation), nowadays accepted if and only if |x| < 1:
1
1+x
=
1
X
i=0
(x)
i
=1 x + x
2
x
3
+ ...
From x =1(of course this is not correct)
5
we should have: 1 1+1 1+
···=
1
2
. Gottfried Wilhelm Leibniz (1646–1716) studied Grandi’s series and in his
Normat 4/2005 Giorgio T. Bagni 175
letter to Jacopo Riccati (1676–1754) probably written in 1715, stated that Grandi’s
solution is correct (“In Acta Eruditorum Lipsiae I think I have solved this problem”:
Michieli, 1943, p. 579).
6
Moreover, Leibniz studied Grandi’s series in some letters
(1713–1716) to the German philosopher Christian Wolf (1678–1754)
7
, where he
introduced the “probabilistic argument” (that influenced, for instance, Johann and
Daniel Bernoulli).
Let us see more precisely the Leibnitian argument dealing with the question
if 1 1+1 1+1 1+&c. in infinitum is
1
2
and how we can avoid the absurdity
that in this statement can be recognised. As a matter of fact, when we consider
an infinity of 1 1=0, it seem impossible to state that the final result can
be
1
2
. (Leibniz, 1716, p. 183)
8
Leibniz noticed that if we “stop” the infinite series 11+11+··· (so we consider
a “series finita”: Leibniz, 1716, p. 187), it is possible to obtain either 0 or 1 with
the same “probability”. As a matter of fact,
the series finita [. . . ] can have an even number of terms, and the final one is
negative: 1 1, or 1 1+1 1, or 1 1+1 1+1 1 [. . . ] or it can
have an odd number of terms, and the final one is positive: 1, or 1 1+1, or
1 1+1 1+1. (Leibniz, 1716, p. 187)
9
The Leibnitian original conclusion is the following:
When numbers’ nature vanishes, our possibility to consider even numbers or odd
numbers vanishes, too. [. . . ] So taking into account what is stated by the authors
that wrote about evaluations, [. . . ] we ought to take the arithmetical average [of
176 Giorgio T. Bagni Normat 4/2005
0 and 1], i.e. the half of their s um; and in this case nature itself respects justitiae
law. (Leibniz, 1716, p. 187)
10
Hence the most probable value is the average of 0 and 1, that is
1
2
. It is worth
highlighting this essential passage from finite cases (series finita) to the final infinite
situation: as a matter of fact there is a clear and important change, and numbers’
nature itself “vanishes”. Would it be possible to relate this perspective to actual
infinity? Of course we do not claim that Leibniz, by that, made explicitly reference
to this conception. However his approach is interesting and may deserve further
historical analysis. Finally, Leibniz (1716, p. 188) conceded that “his argument was
more metaphysical than mathematical, but went on to say that there was more
metaphysical truth in mathematics than was generally recognized” (Kline, 1972,
p. 446).
11
Varignon and Riccati
Lagrange and Poiss on also ac-
cepted the previous argument;
but Pierre de Varignon (1654–
1722) noticed that in order to
state:
1
a + b
=
1
a
b
a
2
+
b
2
a
3
···
the condition b<ais
needed (Loria 1929–1933, p.
673), while the convergence of
Grandi’s series to
1
2
can be
obtained by a = b =1. So
Varignon’s note can be inter-
preted as a first consideration
of the role of convergence. Ja-
copo Riccati criticised the con-
vergence of Grandi’s series to
1
2
; in Saggio intorno al sistema
dell’universo (1754), he wrote:
[Grandi’s] argument is interesting, but wrong because it causes contradictions.
[. . . ] Let us consider
n
(1+1)
and, by the common procedure, build n n + n
n etc. =
n
(1+1)
. If it is remembered that 1 1=n n, or 1+n = n +1, we have,
in both series [in this series and in Grandi’s], that there is the same quantity of
zeroes. (Riccati, 1761, I, p. 87)
12
Riccati’s argument deserves a brief remark; he writes
1
2
=11+11+···, “by the
common procedure”, then he introduces the infinite series: n/2=nn+nn+···
Normat 4/2005 Giorgio T. Bagni 177
Let us compare the considered series; we can write:
s =1 1+1 1+1 1+···
=(1 1) + (1 1) + (1 1) + ···=0+0+0+···
s
0
= n n + n n + n n + ···
=(n n)+(n n)+(n n)+··· =0+0+0+···
Through this, Riccati concludes that Grandi’s procedure is incorrect. Of course,
nowadays, this argument cannot be accepted (it is based upon the “common pro-
cedure” referred to indeterminate series); however, Ric cati’s conclusion is correct:
13
The mistake is caused by the use of a series [. . . ] from which it is impossible to
get any c onclusion. In fact, [. . . ] it does not happen that the following terms can
be neglected in comparison with preceding terms; this property is verified only
for convergent series. (Riccati, 1761, I, p. 87)
14
A brief experimental survey
The educational use of historical references must be carefully controlled: as a mat-
ter of fact, the consideration of infinite series can cause inconsistencies in students’
minds: for instance, if a pupil considers an infinite series as an arithmetical oper-
ation, the absence of the s um of Grandis series can cause many doubts (and the
influence of the didactical contract can be highlighted: Sarrazy, 1995). We shall
briefly consider pupils’ opinions regarding Grandi’s series. A test (B agni, 2005)
was proposed to s tudents of two third-year Liceo Scientifico c lasse s, total 45 pupils
(aged 16–17 years), and of two fourth-year Liceo scientifico class, 43 pupils (aged
17–18 years; total: 88 pupils), in Treviso (Italy). Their mathematical curricula were
traditional: in all class es , at the moment of the test, pupils did not know infinite
series; they knew the concept of infinite set.
15
We ask our students to consider
“1 1+1 1+···” (studied “in 1703” by “the mathematician Guido Grandi”),
taking into account that “addends, infinitely many, are always +1 and -1” and to
express their “opinion about it” (time: 10 minutes; no books or calculators allowed):
Answers: the result is 0 26 29%
the result is 1 3 4%
the result can be either 0 or 1 18 20%
the result is
1
2
4 5%
the result is infinite 2 2%
the result does not exist 5 6%
no answer 30 34%
It is worth highlighting that the greater part of the pupils interpreted this ques-
tion as an implicit request to calculate the “sum” of the considered infinite series.
Only 5 students (6%) stated that it is impossible to calculate the sum of Grandi’s
series (and their answers were not provided with clear justifications); it should be
178 Giorgio T. Bagni Normat 4/2005
remembered that 18 pupils, 20%, gave two “results”; 35 pupils (40%) gave a “re-
sult” (a finite or an infinite one) and many pupils gave no answer (34%). Several
students justified their answers in some interviews. Concerning pupils that stated
that the sum of the infinite series 1 1+1 1+··· is 0, some of them made
reference to an argument by Grandi, quoted by Riccati, too (“If I always want to
add 1 and 1, I can write (1 1) + (1 1) and so I can couple 1 and 1: so I am
going to add infinitely many 0: I obtain 0”: Marco, third year, and 15 other pupils).
Students that stated that the sum of the considered se ries is
1
2
made reference to
justifications similar to the argument by Leibniz-Wolf (for instance: “If I add the
numbers I have 1, 0, 1, 0 and always 1 and 0. The average is
1
2
”: Mirko, fourth
year). Audio-recorded material and transcriptions allowed us to point out a salient
short passage (1 minute and 35 seconds, 9 utterances):
[1] Researcher: Why did you write that the result is
1
2
?
16
[2] Mirko: Oh, well, I start with 1, so I have 0, then 1, 0 and so on. There are
infinitely many +1 and 1.
17
[3] Researcher: That’s true, but how can you say
1
2
?
18
[4] Mirko: If I add the numbers, I obtain 1, 0, 1, 0 and always 1 and 0. The
average is
1
2
.
19
[5] Researcher: And so?
20
[6] Mirko: The numbers that I find are 1, 0, and 1, 0, and 1, 0 and so on: clearly,
every two numbers, one of them is 0 and one of them is 1. So these possibilities
are equivalent and their average is
1
2
.
21
[7] Mirko: [after 12 seconds] Perhaps my answer is strange, or wrong, but I don’t
see a different correct result: surely both the results 0 and 1 are wrong. If I say
that the result is one of that numbers, for instance 1, I forget all the other
numbers, an infinite sequence of 0.
22
[8] Researcher: So in your opinion both 0 and 1 cannot be considered the correct
answer.
23
[9] Mirko: Alright, and in this case what is the result? I wrote that
1
2
is the
results of the operation because
1
2
is the average, so it is a number that, in a
certain sense, contains both 0 and 1.
24
Mirko stated that “every two numbers, one of them is 0 and one of them is 1” ([4])
and “the average [. . . ] is a number that, in a certain sense, contains both 0 and 1”
([9]). So he did not make explicit reference to the probability: he mainly tried to
find a result for the considered problem, and this is an educational issue (influenced
by the didactical contract); in the 18th century, the probabilistic argument was
based upon a slightly different remark, according to which if we “stop” the infinite
series 1 1+1 1+···, it is possible to obtain both 0 and 1 with the same
“probability.”
25
So really students’ justifications are sometimes similar to some
examples from the history of mathematics (Furinghetti & Radford, 2002); but our
present cultural context is quite different from the context that characterised the
historical reference.
Normat 4/2005 Giorgio T. Bagni 179
Mirco’s protocol
With regard to the notion of infinity (potential infinity and actual infinity), it is
interesting to examine Mirko’s protoc ol.
Mirko’s use of visual elements is interesting: in particular, it is worth noting the
horizontal sign by which Mirko divides the “finite” from the “infinite”. After that
sign, he writes “infinity” (“infinito”). This protocol reveals a clear difference, in
Mirko’s mind, between those situations: when we consider the first group of steps
(referred to the se ries finita, so to speak), we have an alternating sequence of num-
bers, both 0 and 1 (and this sequence is highlighted by segments connecting the
numbers). But the final situation (“infinity”) is completely different: now we have
no place enough for “two” numbers (0, 1), so we must write only one value after
the arrows, i.e. the result: their average,
1
2
. Of course Mirko’s protocol does not
evoke a potential conception of infinity: one would think his visualization suggests
that infinity is not just a continuous process that can be indefinitely lengthened.
More precisely, infinity is referred to a single “entity”, a single “place” after the
arrows and the marked line. Nevertheless this situation is not enough to consider
Mirko’s approach with reference to actual infinity, but we can point out an ideal
and interesting analogy with the aforementioned ancient Leibnitian solution. How-
ever the main point to be discussed is the following: c an we state that historical
aspect, in particular Leibniz-Wolf’s probabilistic argument, is really essential in
order to suggest a conception grounded on actual infinity? More properly, in our
opinion, the crucial as pect is educational: the arrows, in Mirko’s protocol, lead to
the final “result of the operation”, whose primary importance is emphasized by the
didactical contract: the role of Leibniz-Wolf’s probabilistic argument is minor, in
this step. So we can state that considered elements ought to be connected in the
following order:
180 Giorgio T. Bagni Normat 4/2005
Guido Grandi’s series is a problem to be solved
(and)
A problem is solved if and only if its correct result is provided [didactical con-
tract]
(so)
Mirko has to pass from the two “partial results” 0 and 1 to the correct result
(one and only one [didactical contract])? actual infinity?
(so)
Being 0 and 1 seemingly equivalent, Mirko calculates their average (“a number
that, in a certain sense, contains both 0 and 1”): by that, he respects “justitiae
law” (Leibniz, 1716, p. 187)
So the role of aforementioned historical references is really interesting, but it is
strictly connected to educational aspects: and this leads us to state that Mirko’s
and Leibniz-Wolf’s arguments are hardly referred to the “same” epistemological
obstacle.
Concluding remarks
In our opinion some examples from the history do help with the introduction of
an important topic of the mathematical curriculum of High School.
26
Nevertheless,
historical e xamples clearly stimulated many pupils, but an explicit institutionali-
sation by the teacher is clearly necessary. In order to conclude our reflections, we
turn back to the theoretical framework mentioned at the beginning of this paper.
Is it p oss ible to state, and to use in educational practice, a paralleling of history
with learning processes? According to several researchers (e.g. Sfard, 1991, p. 10),
the historical development of a concept can be regarded as the sequence of steps.
In the early step the focus is mainly operational; the structural point of view is
not a primary one: for example, as previously noticed, concerning infinite series, in
this early step main questions of convergence were not considered (for instance, let
us remember once again Riccati’s aforementioned argument). A similar situation
can be pointed out from the cognitive point of view: of course, in the early step
pupils approach concepts mainly by intuition, without a full comprehension of the
matter; then the learning becomes better and better. Some experimental results
seem to suggest that in the educational passage from the early step to the mature
one we can see, in pupils’ minds, reactions and doubts that we can find in the
passage from the early step to the mature one as regards the savoir savant (Tall
& Vinner, 1981).
27
But several issues ought to be considered: for instance, what
do we mean by “pupil minds”? More generally, can we still consider our mind as a
“mirror of nature” (Rorty, 1979) and make reference to our “inner representations”
uncritically? According to W. V. O. Quine,
epistemology, or something like it, simply falls into place as a chapter of psy-
chology and hence of natural science. It studies a natural phenomenon, viz., a
particular human subject. (Quine, 1969, p. 82)
Moreover R. Rorty underlines the crucial importance of “the community as source
of epistemic authority” (Rorty, 1979, p. 380), and states:
Normat 4/2005 Giorgio T. Bagni 181
We need to turn outward rather then inward, toward the social context of jus-
tification rather than to the relations between inner representations.
(Rorty, 1979, p. 424)
Of course this theoretical perspective needs further research in order to be effec-
tively applied in mathematics education. Nevertheless, we can state that a soci-
ological approach is very important, and points out some difficulties (Bagni &
D’Amore, 2005) from the educational viewp oint: since an operational conception
can be considered before a structural one, as far as infinite series is concerned the
passage from an operational conception to a structural one has been arduous, be-
cause of the necessity of some basic notions, like the limit concept,
28
which was not
considered in many particular cultural contexts. In our opinion a crude paralleling
of history with learning processes would connect two cultures referring to quite
different contexts (Radford, 1997), so it cannot be used without a consideration of
the social and cultural backgrounds.
29
We can conclude that the introduction of in-
finite series in the classroom is not simple and several aspects can be considered.
30
The examined example is meaningful: we cannot forget that Mirko made reference
to Leibniz-Wolf’s probabilistic argument, and his behaviour can be considered in
the sense of a first approach to actual infinity; but it is important to highlight that
his choice is essentially related to the clause of didactical contract that emphasizes
the importance of the result of a given problem or operation. The main problem
of the passage from finite to infinite is a cultural one, and historical issues are
important in order to approach it, although, undoubtedly, the historical approach
can be considered together with other (educational) approaches.
31
So let us finally
quote T. Heiede, who states:
The history of mathematics is not just a box of paints with which one can make
the picture of mathematics more colourful, to catch interest of students at their
different levels of education; it is a part of the picture itself. If it is such an
important part that it will give a better understanding of what mathematics is
all about, if it will widen horizons of learners, maybe not only their mathematical
horizons [. . . ] then it must be included in te aching. (Heiede, 1996, p. 241)
Acknowledgements
The author would like to thank Torkil Heiede for the valuable help and for many
important bibliographical references.
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Notes
1
Of course it is possible to employ several visual representations (see for instance: Duval,
1995; Bagni, forthcoming-a and forthcoming-b).
2
In fact, Tacquet made reference to ancient mathematics without any historical contex-
tualization. His position, too, must be contextualised: we cannot suppose the presence
of our philosophical awareness in the 17th century (of course it is necessary to take into
account both the period in which a work was written and the period of its edition or
comment: Barbin, 1994; Dauben & Scriba, 2002).
3
Some experimental data discussed in this paper are reprised by: Bagni, 2005.
4
Euler and Fourier also thought that 1 1+1 1+··· =
1
2
.
5
From an educational point of view, it can be noticed that this (wrong) result can
be achieved by the (wrong) procedure: from s =1 1+1 1+··· we should have:
s =1 (1 1+1+···) and s =1 s, so s =
1
2
. Of course, nowadays, this procedure
cannot be accepted: it is clearly based upon a quite incorrect use of arithmetical properties
and upon the implicit statement that 1 1+1 1+··· is a number s, and we know that
this is false (Bagni, 2005).
6
In this paper the translations are ours.
7
Leibnitian letters to Wolf were published in Acta Eruditorum Lipsiae, Tom. V. ab an.
1711 ad an. 1719 Epist. G. G. L. ad V. claris. Ch. Wolfium. It is worth noting that Leibniz
corresponded with most of the scholars in Europ e: as a matter of fact he had over 600
correspondents (see for instance his Commercium Phisosphicum et mathematicum with
Johann Bernoulli: Lebniz & Bernoulli, 1745).
184 Giorgio T. Bagni Normat 4/2005
8
“Utrum 1 1+1 1+1 1+&c. in infinitum sit
1
2
; & quomodo absurditas evitari possit,
quae in tali enuntiatione se ostendere videtur. Nam cum infinities o ccurrere videatur
1 1=0, non apparet quomodo est veris nihilis infinities repetitis possit fieri
1
2
” (Leibniz,
1716, p. 183).
9
“Series finita [. . . ] vel enim constat ex numero membrorum pari & terminatur per veluti:
1 1, aut 1 1+1 1, aut 1 1+1 1+1 1 [. . . ] vel numero membrorum impari, &
terminatur per +, veluti: 1, aut 1 1+1, aut 1 1+1 1+1” (Leibniz, 1716, p. 187).
10
“Tunc evanescente natura numeri, evanescit etiam paris aut imparis assignabilitas [. . . ].
Et quoniam ab iis qui de aestimatione scripsere, [. . . ] sumi debere medium Arithmeticum,
quod est dimidium summae; itaque natura rerum eandem hic observat justitiae legem”
(Leibniz, 1716, p. 187).
11
“Porro hoc argumentandi genus, etsi Metaphysicum magis quam Mathematicum videa-
tur, tamen firmum est: & aliorum Canonum Verae Metaphysicae (quae ultra vocabulorum
nomenclaturas procedit) major est usus in Mathesi, in Analysi, in ipsa Geometria, quam
vulgo putatur” (Leibniz, 1716, p. 188).
12
“Quanto il discorso è ingegnoso, altrettanto è fallace, perché si tira dietro delle insan-
abili contraddizioni. [. . . ] Assunta la frazione n/(1 + 1), col solito metodo ne formo la
serie n n + n n etc. = n/(1 + 1). E giacché si verifica l’equazione 1 1=n n, o sia
1+n = n +1, ne segue che prorogate del pari all’infinito amendue le progressioni [. . . ],
tanti nulla né più né meno conterrà la prima quanti la seconda”(Riccati, 1761, I, p. 87).
13
Riccati’s statement can be related to ideas that mathematicians were going to point
out in the 18th century; finally, in Disquisitiones generales circa seriem infinitam, Gauss
considered the notion of convergence correctly.
14
“Il paralogismo consiste in ciò, che il lodato Scrittore ha fatto uso d’una serie [. . . ]
dalle quali, come altresì dalle divergenti, nulla ci vien fatto di conchiudere. E la ragione
si è, che [. . . ] non succede mai, che i termini susseguenti possano trascurarsi, siccome
incomparabili con gli antecedenti; la qual proprietà alle solo serie convergenti si compete”
(Riccati, 1761, I, p. 87).
15
The researcher was not the mathematics teacher of the pupils, however, he was present
in the classroom with the teacher and the pupils; the experience took place during a lesson
in the classroom.
16
“Perché hai scritto che il risultato è
1
2
?”
17
“Beh, all’inizio cè 1, poi fa 0, poi 1, o e via. Ci sono infiniti +1 e 1.”
18
“Vero, ma perché
1
2
?”
19
“Se faccio le somme ottengo 1, 0, 1, 0 e sempre 1 e 0. La media è
1
2
.”
20
“E allora?”
21
“I numeri che si trovano sono 1, 0, e 1, 0, e 1, 0, sempre così: è ovvio, ogni due numeri
uno è uno 0 e l’altro è un 1. C’è la stessa possibilità e la media fa
1
2
.”
22
“Magari il mio è un discorso strano, magari anche sbagliato, ma non riesco a fare una
cosa diversa: 0 e 1 non vanno bene di sicuro. Se dico che il risultato è uno di quelli, tipo
1, non conto tutti gli altri numeri, tutta la infinita fila di 0.”
23
“Dunque tu dici che 0 e 1 non sono risultati giusti.”
24
“Va bé e allora qual è il risultato? Io lì ho mess o
1
2
come risultato dell’operazione
perché
1
2
è la media, cioè quel numero che in un certo senso contiene 0 e 1.”
25
Let us remember the importance of the probability in the mathematical researches in
the 18th century (for instance, in Acta Eruditorum 1682–1716, we find either the quoted
Leibnitian letter to Wolf or Bernoulli’s Specimina Artis Conjectandi).
26
Of course in this research the considered sample is rather small; it would be necessary
to identify sampling criteria and pre-course intuitions: so we cannot give general results.
27
Since processes of teaching-learning take place nowadays, the transposition didactique
can also be based upon the results achieved in the mature step of the development of
Normat 4/2005 Giorgio T. Bagni 185
the mathematical knowledge; with regard to teachers, of course, a relevant historical-
phislosophical skill is needed.
28
This has relevant conse quences for education. For instance, some remarks in the his-
torical part can suggest the following issue: is it reasonable to introduce convergence in
schools without prior introduction of the limit notion, since this is what happened histor-
ically? (Bagni, 2005).
29
Let us moreover notice that frequently a better use of the developed systematic struc-
ture of mathematics for teaching runs counter to the direct paralleling of history with
learning processes.
30
For instance, embodiment (Lakoff & Núñez, 2000) is one of the most important issues
of research into mathematics education and it is relevant to investigate further connections
between perceptions and symbols. However the fundamental work by Lakoff and Núñez
is devoted to cognitive aspects: the crucial point is the passage from finite to infinite;
and metaphorical reasoning, clearly very important from the educational point of view,
must be controlled by the teacher in order to avoid dangerous misguided generalisations
(Bagni, 2000b).
31
Several questions are still open: for instance, the reading of primary sources can be an
important tool (Fauvel & van Maanen, 2000), but it needs a clear consideration of the
historical evolution of representative registers (Bagni, forthcoming-a).
